https://file.aiquickdraw.com/mathbot/file/1728290318309rn0ge556.jpg

The problem provides the following angle measures:

- \( m \angle LKQ = 3x + 11 \)
- \( m \angle LKJ = 21x - 3 \)
- \( m \angle QKJ = 130^\circ \)

From the diagram (and typical geometric notation), we can assume that these angles form a triangle, where the sum of the angles in any triangle is \(180^\circ\).

Therefore, we can write the equation:

\[
m \angle LKQ + m \angle LKJ + m \angle QKJ = 180^\circ
\]

Substituting the given values:

\[
(3x + 11) + (21x - 3) + 130 = 180
\]

Now, we can simplify and solve for \(x\):

1. Combine like terms:
\[
3x + 21x + 11 - 3 + 130 = 180
\]
\[
24x + 138 = 180
\]

2. Subtract 138 from both sides:
\[
24x = 42
\]

3. Divide both sides by 24:
\[
x = \frac{42}{24} = \frac{7}{4} = 1.75
\]

So, \( x = 1.75 \).

Let me know if you need more details or have any questions!

### Follow-up questions:
1. What is the measure of \( m \angle LKQ \) when \( x = 1.75 \)?
2. What is the measure of \( m \angle LKJ \) when \( x = 1.75 \)?
3. How can you verify if the angle measures add up to \(180^\circ\)?
4. Can this method apply to any triangle with given expressions for the angles?
5. What would happen if the sum of the angles exceeded \(180^\circ\)?

### Tip:
Always check that your final angle values add up to \(180^\circ\) in any triangle problem to ensure correctness!

m∠LKQ = 3x + 11, m∠LKJ = 21x - 3, and m∠QKJ = 130°. Find x.

Learn how to find the value of x in a triangle given the angle measures: m∠LKQ = 3x + 11, m∠LKJ = 21x - 3, and m∠QKJ = 130°. This problem uses algebra and the triangle angle sum theorem to solve for x, suitable for students in Grades 8-10.

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